Domain wall dynamics of periodic magnetic domain patterns in Co2MnGe-Heusler microstripes

Highly symmetric periodic domain patterns were obtained in Co2MnGe-Heusler microstripes as a result of the competition between growth-induced in-plane magnetic anisotropy and shape anisotropy. Zero field magnetic configurations and magnetic field-induced domain wall (DW) motion were studied by magnetic force microscopy-image technique for two different cases: dominant uniaxial- and dominant cubic in-plane anisotropy. We implemented a magneto-optical Kerr effect susceptometer to investigate the DW dynamics of periodic domain structures by measuring the in-phase and out-of-phase components of the Kerr signal as a function of magnetic field frequency and amplitude. The DW dynamics for fields applied transversally to the long stripe axis was found to be dominated by viscous slide motion. We used the inherent symmetry/periodicity properties of the magnetic domain structure to fit the experimental results with a theoretical model allowing to extract the DW mobility for the case of transverse DWs (μTDW = 1.1 m s−1 Oe−1) as well as for vortex-like DWs (μVDW = 8.7 m s−1 Oe−1). Internal spin structure transformations may cause a reduction of DW mobility in TDWs as observed by OMMFF simulations.


Introduction
Domain wall (DW) propagation in laterally patterned magnetic thin films holds promise for both fundamental interest and potential for technological applications. Fundamentally, DW motion can be induced by external magnetic fields or by spin polarized currents [1][2][3][4][5][6][7][8]. A significant part of measurements with injected spin polarized current have been done on systems with the magnetization direction in-plane. In nanostripes with in-plane magnetization, two DW types have been identified: in thin and narrow stripes, transverse walls (TDWs) are energetically favored, while in thicker and wider stripes vortex-like walls (VDWs) have lower energy [9,10]. The advantage of these systems are broad DWs which are less sensitive to pinning. Spin-transfer torque driven DWs in nanosized permalloy (Py=Ni 80 Fe 20 )-wires nowadays are the basis for the racetrack memory concept development at IBM's Almaden Research Center [11]. However, very high spin-current densities are still required to induce a DW displacement. Py has been the material of choice in the scientific community for the investigation of DW propagation, as it possesses negligible magnetocrystalline anisotropy and low damping. However, materials with low damping are more sensitive to undergo DW spin structure modifications, which can have marked effects on the DW dynamics since both DW velocity and DW pinning are sensitive to the DW spin structure [6,8].
A key parameter for characterizing magnetic DW motion is the DW mobility (μ DW ), which describes the rate of change of wall velocity (v DW ) with variation of the external field. Schryer andWalker [12] derived a 1D model from the Landau-Lifshitz-Gilbert equation, which has been the basis for subsequent additional works [13,14]. These models describe the fundamental aspects of DW dynamics with three characteristic regimes: at low fields, the equation of motion give an exact stationary-state solution characterized by a linear dependence of the velocity on the applied field. In this case the mobility is given by μ DW =γΔ/α, where γ is the gyromagnetic Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. ratio, Δ is the DW width and α is the Gilbert damping parameter. This solution exists only up to a critical field H W denominated as the Walker breakdown field. Above H W , DW movement becomes oscillatory and the wall undergoes changes in its internal spin structure. In this regime, the net average wall velocity first decreases with increasing H, until, for H?H W the net motion results again in an asymptotically linear H-v DW function, but now with a mobility given by μ DW =γΔ/(α+α −1 ). Based on this formalism, Kataja and van Dijken [15] proposed a model considering an alternating (ac) driving field H=H 0 cosωt. Under ac-field excitation DW creep and slide modes can be activated in dependence on the amplitude and frequency of the field. The model proposed in [15] describes the complex Kerr signal (V Kerr =X−iY) as a function of field amplitude (H 0 ) and driving frequency (f=ω/2π) to describe the DW slide regime of regular magnetic domain patterns. This model allows to obtain DW mobility and depinning fields.
Ferromagnetic, half-metal Co-based Heusler alloys like Co 2 MnGe, have been shown to be important materials for spintronic devices [16,17]. Besides the high Curie temperature (T C ∼905 K), their relevance lies mainly on the theoretically predicted 100% spin polarization of the conduction electrons at the Fermi level [18][19][20]; they are also characterized by a low value of the Gilbert damping parameter [21]. Co 2 MnGe has a cubic crystal structure and in the bulk exhibits a weak cubic magnetic anisotropy [22]. When grown as thin film it often exhibits a growth-induced moderate uniaxial magnetic anisotropy (K U ) superimposed on an anisotropy with cubic symmetry (K 4 ) [23 -28]. By an adequate choice of thickness, substrate and deposition temperature, thin films with a dominant uniaxial anisotropy or dominant cubic anisotropy can be obtained [26][27][28][29]. The moderate in-plane anisotropy allows for the formation of well-defined periodic domain patterns by the proper combination with the shape anisotropy, while low damping favors the DW motion in such domain structures.
In most experimental studies the DW motion is analyzed using magnetic field or current pulses, while the DW dynamic response to alternating magnetic fields has rarely been explored. Likewise, there are few reports on magnetic domain structures, magnetotransport and DW propagation in patterned Heusler thin films considering the interplay between magnetocrystalline and shape anisotropies [30][31][32][33]. These results have shown the potential of the Heusler alloys for applications to spintronic devices. Therefore, in the present work, we analyze the DW dynamic response of periodic domain structures in Co 2 MnGe-Heusler microstripes using a magneto-optical Kerr effect susceptometer; since, to the best of our knowledge, experimental reports on DW dynamics in patterned Heusler compounds have not been reported in the literature so far.

Experimental
Arrays containing 510×90 elements of flat rectangular stripes of width w=2.5 μm and length L=20 μm were fabricated from 60 nm thick textured Co 2 MnGe thin films. The films were prepared by rf-sputtering from a Heusler alloy target with stoichiometric composition on a-plane Al 2 O 3 -substrates. Prior to the Co 2 MnGe film growth and in order to induce a high-quality (110) textured growth of the Heusler phase, a 2 nm thick vanadium seed layer was deposited. Films were finally protected against oxidation with a 5 nm thick Al 2 O 3 capping layer. Magnetic investigations of the films were carried out by using magneto optical Kerr effect (MOKE) and superconducting quantum interference device magnetometer. Longitudinal MOKE hysteresis loops were taken as a function of the in-plane azimuthal angle j of the applied magnetic field, where j=0 is parallel to the inplane c-direction of the a-plane Al 2 O 3 substrate. The stripe arrays were fabricated using electron-beam lithography, lift-off techniques and ion beam etching. The magnetic microstructure was investigated at room temperature by magnetic force microscopy (MFM) at a lift scan height of 100 nm by using a multi-mode microscope (NT-MDT) and commercial Co-Cr MFM tips.
The dynamic responses of magnetic DWs was studied by recording the in-phase X and out-of-phase Y components as a function of field amplitude H 0 and driving frequency f by using an ac-MOKE susceptometer. This is an extension of a standard MOKE setup consisting of a diode-laser, polarizing and analyzing optics (Glan-Thompson prisms), a high-speed silicon photo-diode, and lock-in amplifier [34]. The ac magnetic field is generated by Helmholtz coils. A second lock-in amplifier is used as a generator providing a maximum field amplitude of 8.6 mT with driving frequency varying from 0.5 Hz to 150 Hz. The Kerr signal (V Kerr ) is measured by lock-in amplification of the photo-diode output.

Results and discussion
3.1. Thin films: angular dependence of the coercivity Prior to lateral patterning, the Co 2 MnGe thin films were magnetically characterized by rotational MOKE technique to determine the in-plane magnetic anisotropy. As previously reported, the effective in-plane magnetic anisotropy for Co 2 MnGe thin films grown on a-plane Al 2 O 3 can be described by a superposition of a fourfold and a uniaxial anisotropy term [29]. The magnitude of the twofold and fourfold anisotropy constants depends on the growth conditions and can vary appreciably. The microscopic origin for these variations is presently not clear. However, the large scattering of the anisotropy parameters is a characteristic feature of Heusler thin films [17]. The substrate miscut and the specific substrate surface conditions have an important influence.
In the present work, we focus on two different anisotropy properties represented by two samples denominated A and B. These two samples are distinguished by their in-plane magneto-crystalline anisotropies, which are controlled by the growth conditions, in particular by the growth temperature and magnetic fieldassisted deposition. Figures 1(a) and (b) show the polar plots of the coercive field H C (j), measured as a function of the azimuthal angle j at room temperature of samples A and B, respectively. Regarding sample A, a two-fold symmetry can clearly be recognized confirming the dominant uniaxial nature of the in-plane magnetic anisotropy with an easy axis at j=0, which is aligned with the c-axis of the Al 2 O 3 substrate. The magnetocrystalline anisotropy constant K U given by ½(H U M S ) was determined to be 4.1×10 3 J m −3 , considering a saturation magnetization M S =6.3×10 5 A m −1 and an anisotropy field H U =13 mT as obtained by magnetization measurements with the field applied perpendicular to the c-axis. By contrast, the polar plot of the coercive field of sample B evidences a four-fold symmetry of the in-plane anisotropy with two easy axes aligned parallel and perpendicular to the c-axis of Al 2 O 3 . For the latter case, the strength of the fourfold in-plane anisotropy is estimated to be K 4 =2.7×10 3 J m −3 .
We used these two different symmetries of the in-plane anisotropy for the study of magnetic domain structures in microstripes. In the following, we present both equilibrium domain structures and dc-magnetic field-induced DW motion for the lateral patterned samples A and B.

Domain structure in microstripes
By means of e-beam lithography, arrays of stripes were fabricated with a width w=2.5 μm and lengths L=20 μm oriented perpendicular to the magnetic easy axis. Distances between the stripes are 5 μm and 10 μm along the short-and long-axis of the stripes, respectively; which are large enough to suppress magnetostatic interactions between stripes. Figures 2(a) and (b) present MFM images and simulated domain structure (using the object oriented micromagnetic simulation package OOMMF [35] of individual stripes prepared from samples A and B, respectively; the arrows indicate the magnetization directions. Prior to recording the images at zero field, the stripes were magnetically saturated along the magnetic easy axis. In the case of the sample A with dominant in-plane uniaxial anisotropy, competing magnetostatic, magnetocrystalline and exchange interactions result in a regularly symmetric stripe domain configuration with a well-defined domain size [29]. Domains with the magnetization direction perpendicular to the stripe long axis are mostly separated by 180°t ransverse DWs. The domain size δ parallel to the stripe long axis is highly regular and has an average value of δ TDW =1.8±0.2 μm. In contrast, in case of microstripes from sample B, the dominant cubic anisotropy favors the magnetization to lie along as well as perpendicular to the long axis of the stripes; as a result, the magnetization is organized into a diamond-type domain structure as observed in the MFM contrast and OMMF simulations shown in figure 2(b). This system is termed as containing vortex-like DWs or VDWs. The lateral domain length along the stripe axis is δ VDW =2.3±0.2 μm.

dc-magnetic field-induced DW motion 3.3.1. Stripe domain structure (sample A)
When a dc magnetic field is applied in the direction transverse to the stripe long axis, the stripe domain structure (in figure 2(a)) responds as illustrated by the MFM images in figure 3(a): the principal effect of applying the field in that direction is to initiate a DW motion in the direction perpendicular to the field. The sideways DW displacement depends on the field strength; by increasing the field amplitude progressively from H 1 =2.5 mT to H 2 =4.5 mT, DWs displace further (label II). However, this motion may occur only until the maximum distance D/2 is reached, where D is the domain size. Once the magnetic field reaches the annihilation field (μ 0 H an =6.4 mT), DWs synchronously disappear and the magnetization approaches saturation. In magnetic saturation a transverse single domain state is observed (label III). According to the latter, applying a dc field transversally to the stripe long axis leads to both a regular sideways motion of 180°DWs and a synchronic DW annihilation.

Diamond-like domain structure (sample B)
Similar to the stripe domain case above, by applying a field in the direction transverse to the stripe long axis, the diamond-like structure responds by DW motion. A simplified way to describe the DW motion in this case is by an effective translational displacement of the vortex core along the stripe long axis, as schematically shown in figure 3(b) for the fields H 1 =1.2 mT and H 2 =2.0 mT (label II). The sideways vortex core displacement is also limited by D/2; once this length is reached, adjacent domains coalesce and the magnetization approaches saturation (label-III). From static MOKE hysteresis loop measurements (not shown) and MFM observations, the annihilation field was obtained as μ 0 H an =3.2 mT. figure 2 were investigated by applying an ac-magnetic field in the same direction as for the dc-magnetic field (transverse to the stripe long axis). According to the results reported in section 3.3 it can be expected that the domain structures, when exposed to an ac-field H(H 0 ; f), will exhibit an alternating sideways DW motion along the stripe long-axis (see the animations of MFM images in supplementary videos for TDWs and VDWs cases). In this case, the amplitude of the DW displacement depends on both f and H 0 . The implemented ac-MOKE susceptometer allows for recording the in-phase (X) and out-ofphase (Y) components of the Kerr signal as a function of f and H 0 . MOKE experiments were carried out at room temperature by varying the frequency in a range of 0.5 Hz<f< 100 Hz and the amplitude in a range of 0.4 mT<μ 0 H 0 < 8.8 mT. For sideways DW slide motion in a regular magnetic stripe pattern, Kataja and van Dijken in [15] derived analytical expressions (see equation (1)) to describe the normalized X n and Y n components as a function of the frequency f and field amplitude H 0 as:

DW dynamics of periodic magnetic domain structures DW dynamics of the periodic domain structures shown in
According to these expressions, while the in-phase component in the slide regime vanishes, the Y ncomponent shows a dependence on both amplitude and frequency of the applied field. Moreover, it contains terms combining the depinning field H d and the DW mobility μ DW . Experimentally, we first measured the frequency dependence of X and Y for a constant value of H 0 . Figure 4 shows the X and Y components of the measured Kerr signal versus frequency at μ 0 H 0 =4.0 mT for stripe array of sample A. The response of the system for this field amplitude can be divided into three dynamic regimes: at high driving frequencies, f>70 Hz, no MOKE signal is detected, i.e. both X and Y components nearly vanish; in this frequency regime DWs remain pinned. As soon as the frequency reaches a value around 70 Hz, the out-of-phase component (Y n ) starts to increase, marking the depinning frequency f d at which the oscillatory and viscous slide motion of the existing DWs sets in. With decreasing frequency the amplitude of wall displacement grows, thus the imaginary Kerr signal increases, reaching a maximum at the peak frequency f p =36 Hz. This peak frequency value marks the point when the walls have enough time to reach a displacement equal to half of the domain width (D/2). At this point domains coalesce and the magnetization reaches saturation. Below this frequency, the system alternates between the two opposite saturated states giving rise to the switching regime.
The response of the DW system to the variation of the field amplitude can be seen in the inset of figure 4. This figure shows the Y n -component versus frequency measured at different field amplitudes. Note the evolution of the DW response by increasing H 0 from 1.3 to 5 mT, which is characterized by a gradually enhanced signal and a shift of the peak to higher frequencies. Consequently, the larger H 0 , the higher the frequency peak which marks the crossover from DW slide to the switching regime.
Following equation (1), we plot the experimental Y n -component as a function of 1/f in order to extract T H , 0 ( ) which contains information about wall mobility and depinning fields. The solid lines in this plot are fits to the experimental data using equation (1). The orange solid line in this graph is a fit to the experimental data for the TDW-system; we thus obtain μ 0 H d =0.47 mT, and a DW mobility of μ DW =1.1 m s −1 Oe −1 . Following the same procedure for the case of VDW-system (pink solid line), we obtain a depinning field of μ 0 H d =0.98 mT and a DW mobility of μ DW =8.7 m s −1 Oe −1 . Similar values for the DW mobility have been reported for Ho-doped Py wires [8]. According to our results, the vortex-like DWsystem responds faster to the applied field, exhibing a DW mobility about seven times larger than the TDWsystem.
It is well known that DW dynamics is sensitive to the internal DW structure. The DW internal structure depends on the thickness, width and material parameters of the constituting magnetic nanostructure [9,10,36]. According to analytical models [12,14], the DW mobility depends directly on the DW width; since vortex-like DWs exist for larger widths, one could expect higher mobility for a vortex wall. Moreover, numerical calculations [37] indicate that transverse and vortex walls behave differently. Numerical results of DW mobility as a function of wire widths confirm that the mobility behavior of TDWs approximates the 1D model quite well, while VDWs are characterized by a drastic increase of DW mobility deviating strongly from this theoretical prediction. These analytical as well as numerical calculations support our experimental observation for Co 2 MnGe stripes that VDWs can move faster than TDWs.
One possible reason for the observed low mobility of TDWs compared to the VDWs may be related to changes of the DW internal spin structure as reported in [6] and [8]. In their work the authors deduced a direct correspondence between the magnitude of damping constant α and the rigidity of DWs. For small α the DW   spin structure undergoes periodic distortions that lead to a reduction of the mobility. Motivated by these results, we searched for possible modifications in the DW structure of TDW and performed micromagnetic simulations in order to reveal the evolution of the spin structure of the TDW-system when increasing the external field. Figure 6 shows OOMMF simulations of the spin structure in a stripe of the same dimensions as the experimental one (t=60 nm, w=2.5 μm and L=20 μm). Simulations were perfomed for two different field amplitudes (H 1 and H 2 ) with H 1 <H an and H an >H 2 >H 1 .
Our OOMMF simulations indeed reveal that the transverse DWs are not completely tranversal, but some kind of flux closure structures are formed. When the field is increased from H 1 to H 2 , the DW structure changes from a two-vortex state into a three-vortex state, as highlighted in figure 6. Such kind of internal spin structure transformations may cause the DW mobility to slow down as observed for TDWs.

Conclusions
We have investigated the DW dynamics of periodic domain structures by using ac-MOKE techniques on Co 2 MnGe-Heusler microstripes. The inherent symmetry/periodicity of the domain patterns allows an analysis of the experimental data using existing theoretical models for DW dynamics. The analysis of the in-phase and out-of-phase components of the Kerr signal obtained as a function of magnetic field frequency and amplitude revealed that DW dynamics is dominated by viscous slide DW motion. The quantitative analysis of the DW mobility for two different symmetries i.e. transverse DWs and vortex-like DWs give evidence that the wall mobility for VDWs is larger than for TDWs, in agreement with theoretical predictions. We report for the first time experimental values for the DW mobility in Co 2 MnGe microstructures, and find that they are comparable with experimental values derived for Py. Since a high mobility of DWs is an important and desirable property in many spintronic devices, our results give further support for the potential of ferromagnetic Heusler half metals in the field of spintronics.